91Ó°ÊÓ

A formula for the standard deviation of the energy of a system of N identical harmonic oscillators (such as in an Einstein solid), in the high-temperature limit. Divide by the average energy to obtain a measure of the fractional fluctuation in energy.

In terms of the heat capacity $C_V$, the standard deviation ${\mathrm{σ}}_E$is given by:

${\mathrm{σ}}_E=T\sqrt{C_{v}k}$

Consider a system of N harmonic oscillators operating at high temperatures, where $C_V=Nk$is the heat capacity at constant volume.

$$\begin{gathered} {\mathrm{σ}}_E=T\sqrt{N{k}^2} \\ {\mathrm{σ}}_E=kT\sqrt{N} \end{gathered}$$

The standard deviation divided by the average energy $\stackrel{¯}{E}=NkT$equals the fractional fluctuation in energy, so:

$$\begin{gathered} \frac{{\mathrm{σ}}_E}{\stackrel{¯}{E}}=\frac{kT\sqrt{N}}{NkT} \\ \frac{{\mathrm{σ}}_E}{\stackrel{¯}{E}}=\frac{1}{\sqrt{N}} \end{gathered}$$

The energy fluctuation for N=1

$$\begin{gathered} {\left.\frac{{\mathrm{σ}}_E}{\stackrel{¯}{E}}\right|}_{N=1}=\frac{1}{\sqrt{1}}=1 \\ {\left.\frac{{\mathrm{σ}}_E}{\stackrel{¯}{E}}\right|}_{N=1}=1 \end{gathered}$$

The energy fluctuation for N=10⁴ is:

$$\begin{gathered} {\left.\frac{{\mathrm{σ}}_E}{\stackrel{¯}{E}}\right|}_{N={10}^4}=\frac{1}{\sqrt{{10}^4}}=0.01 \\ {\left.\frac{{\mathrm{σ}}_E}{\stackrel{¯}{E}}\right|}_{N={10}^4}=0.01 \end{gathered}$$

The energy fluctuation for N=10²⁰ is:

$$\begin{gathered} {\left.\frac{{\mathrm{σ}}_E}{\stackrel{¯}{E}}\right|}_{N={10}^{20}}=\frac{1}{\sqrt{{10}^{20}}}=1.0×{10}^{-10} \\ {\left.\frac{{\mathrm{σ}}_E}{\stackrel{¯}{E}}\right|}_{N={10}^{20}}=1.0×{10}^{-10} \end{gathered}$$

Because the energy fluctuation diminishes as the number of subsystems increases, we can utilise the energy E instead of the average energy $\stackrel{¯}{E}$for large N.